theorem
  f is_hpartial_differentiable`21_in z0 implies r(#)pdiff1(f,2)
  is_partial_differentiable_in z0,1 & partdiff((r(#)pdiff1(f,2)),z0,1) = r*
  hpartdiff21(f,z0)
proof
  assume
A1: f is_hpartial_differentiable`21_in z0;
   reconsider r as Real;
A2: pdiff1(f,2) is_partial_differentiable_in z0,1 by Th11,A1;
  partdiff((r(#)pdiff1(f,2)),z0,1) = r*partdiff(pdiff1(f,2),z0,1) by
PDIFF_1:33,A2;
  hence thesis by A1,A2,Th15,PDIFF_1:33;
end;
